Very stable and wobbly loci for elliptic curves

TL;DR

This paper investigates the properties of very stable and wobbly bundles over elliptic curves, proving that the wobbly locus forms a divisor in the moduli space and confirming a conjecture by Drinfeld for genus 1.

Contribution

It establishes that the wobbly locus is always a divisor in the moduli space of semistable bundles on elliptic curves and extends Drinfeld's conjecture to genus 1.

Findings

Wobbly locus is always a divisor in the moduli space.

Twisted stable bundles on elliptic curves are not very stable for positive twists.

Confirmed Drinfeld's conjecture for genus 1.

Abstract

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus $1$. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus $1$ curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.